ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
NAVIER-STOKES EQUATIONS IN THE HALF-SPACE IN VARIABLE EXPONENT SPACES OF CLIFFORD-VALUED
FUNCTIONS
RUI NIU, HONGTAO ZHENG, BINLIN ZHANG Communicated by Vicentiu Radulescu
Abstract. In this article, we study the steady generalized Navier-Stokes equations in a half-space in the setting of variable exponent spaces. We first establish variable exponent spaces of Clifford-valued functions in a half-space.
Then, using this operator theory together with the contraction mapping prin- ciple, we obtain the existence and uniqueness of solutions to the stationary Navier-Stokes equations and Navier-Stokes equations with heat conduction in a half-space under suitable hypotheses.
1. Introduction
Since Kov´aˇcik and R´akosn´ık [24] first studied the spacesLp(x)andWk,p(x), more and more attention are paid to Lebesgue and Sobolev variable exponent spaces and their applications to differential equations. See [7, 8] for basic properties of variable exponent spaces and [21, 33] for recent overviews of differential equations with variable growth. It is well-known that one of the reasons that forced the rapid expansion of the theory of variable exponent function spaces has been the models of electrorheological fluids introduced by Rajagopal and R˚uˇziˇcka [29, 30], which can be described by the boundary-value problem for the generalized Navier-Stokes equations. In the setting of variable exponent spaces, Diening et al. [5] proved the existence and uniqueness of strong and weak solutions of the Stokes system and Poisson equations for bounded domains, the whole-space and the half-space, respectively.
In the previous decades, the study of these spaces has been stimulated by prob- lems in elastic mechanics, calculus of variations and differential equations with variable growth conditions, see [9, 12, 31, 32, 34, 35, 36] and references therein.
As a powerful tool for solving elliptic boundary value problems in the plane, the methods of complex function theory play an important role. One way to extend these ideas to higher dimension is to begin with a generalization of algebraic and geometrical properties of the complex numbers. In this way, Hamilton studied the algebra of quaternion in 1843. Further generalizations were introduced by Clifford
2010Mathematics Subject Classification. 30G35, 35J60, 35Q30, 46E30, 76D03.
Key words and phrases. Clifford analysis; variable exponent; Navier-Stokes equations;
half-space.
c
2017 Texas State University.
Submitted January 28, 2017. Published April 5, 2017.
1
in 1878. He initiated the so-called geometric algebras or Clifford algebras, which are generalizations of the complex numbers, the quaternions, and the exterior algebras, see [19]. After that, Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. Clifford analysis as an active branch of mathematics concerned with the study of Dirac equation or of a generalized Cauchy-Riemann system, in which solutions are de- fined on domains in the Euclidean space and take values in Clifford algebras, see the monograph of Brackx et al. [1]. It is worthy mentioning that G¨urlebeck and Spr¨oßig [14, 15] developed an operator calculus, which is analogous to the known complex analytic approach in the plane and based on three operators: a Cauchy- Riemann-type operator, a Teodorescu transform, and a generalized Cauchy-type integral operator, to investigate elliptic boundary value problems of fluid dynam- ics over bounded and unbounded domains, especially the Navier-Stokes equations and related equations. Of course, there are a number of unsolved basic problems involving the Navier-Stokes equations. This is mainly due to the problem concern- ing the solvability of the corresponding linear Stokes equations over domains, see [2, 16]. As Galdi [20] pointed out, the study of the Stokes problem in the half-space possesses an independent interest and it will be fundamental for the treatment of other linear and nonlinear problems when the region of flow is either an exterior domain or a domain with a suitable unbounded boundary.
On the one hand, Diening et al. [4] studied the following model introduced in [29, 30] to describe motions of electrorheological fluids:
−divM(Du) + (u· ∇)u+∇π=f x∈Ω divu= 0 x∈Ω
u= 0 x∈∂Ω,
(1.1)
where Ω is a bounded domain with Lipschitz boundary∂Ω, f ∈(W01,p(x)(Ω))∗ = W−1,p0(x)(Ω), 2n/(n+ 2) < p− ≤ p+ <∞ and the operator M satisfies certain natural variable growth conditions. The authors obtained the existence of weak solutions in (W01,p(x)(Ω))n×Ls0(Ω), heres:= min
(p+)0, np−/2(n−p−) ifp−< n and s := (p+)0 otherwise, Ls0(Ω) := {π ∈ Ls(Ω) : R
Ωπdx = 0}. Diening et al.
[5] studied the Stokes and Poisson problem in the context of variable exponent spaces in bounded domains and in the whole space. In the half-space case, the authors employed a localization technique to reduce the interior and the boundary regularity to regularity results on the half-space. While it should be pointed out that our attempt is to give a unified approach to deal with physical problems modelled by the generalized Navier–Stokes equations, which is quite different with approaches of some authors, for example, we refer to the monograph [3].
On the other hand, it is natural to focus on theA-Dirac equations if one interests in extending the classical Dirac equations. In [26, 27], Nolder first introduced the general nonlinear A-Dirac equations DA(x, Du) = 0 which arise in the study of many phenomena in physical sciences. Moreover, he developed some tools for the study of weak solutions to nonlinearA-Dirac equations in the spaceW01,p(Ω,C`n).
Inspired by his works, Fu and Zhang in [10, 11] were devoted to the the existence of weak solutions for the general nonlinearA-Dirac equations with variable growth.
For this purpose, the authors established a theory of variable exponent spaces of Clifford-valued functions with applications to homogeneous and non-homogeneous
A-Dirac equations, see also [37]. Recently, Fu et al. [9, 28, 38] established a Hodge- type decomposition of variable exponent Lebesgue spaces of Clifford-valued func- tions with applications to the Stokes equations, the Navier-Stokes equations and theA-Dirac equationsDA(Du) = 0. By using the Hodge-type decomposition and variational methods, Molica Bisci et al. [25] studied the properties of weak solutions to the homogeneous and nonhomogeneousA-Dirac equations with variable growth.
Motivated by the above works, we study of Navier-Stokes equations in a half- space in variable exponent spaces of Clifford-valued functions. To the best of our knowledge, this is the first time to investigate Navier-Stokes equations over un- bounded domains in such spaces. To this end, we first establish variable expo- nent spaces of Clifford-valued functions in the half-space. Then, using an iteration method which requires the solution of a Stokes-problem in every step of iteration, we study the existence and uniqueness of Navier-Stokes equations in a half-space.
There is no doubt that we encounter serious difficulties, for instance, the Sobolev embedding is not compact in a half-space, and operator theory in variable exponent spaces of Clifford-valued functions in a half-space is still unknown. Anyway, our attempt would be a meaningful exploration in the study of fluid dynamics, and the whole treatment applies to a much larger class of elliptic problems.
This article is organized as follows. In Section 2, we start with a brief summary of basic knowledge of Clifford algebras and then investigate basic properties of variable exponent spaces of Clifford-valued functions in a half-space. In Section 3, with the help of the results of Diening et al. [5], we prove the existence and uniqueness of the Stokes equations in the context of variable exponent spaces in a half-space. In Section 4, we present an iterative method for the solution of the stationary Navier- Stokes equations. Using the contraction mapping principle, we prove the existence and uniqueness of solutions to the Navier-Stokes equations in W01,p(x)(RN+,C`n)× Lp(x)(RN+,R) under certain hypotheses. In Section 5, using the contracting mapping principle, we obtain the existence and uniqueness of solutions for the Navier-Stokes problem with heat conduction under some appropriate assumptions.
2. Preliminaries
2.1. Clifford algebras. We first recall some related notions and results concerning Clifford algebras. For a detailed account we refer to [14, 15, 26, 27].
Let C`n be the real universal Clifford algebras overRn. Denote C`n by C`n= span{e0,e1,e2, . . . ,en, e1e2, . . . ,en−1en, . . . ,e1e2· · ·en}
where e0 = 1(the identity element inRn), {e1,e2, . . . ,en}is an orthonormal basis ofRn with the relation eiej+ ejei=−2δije0. Thus the dimension of C`n is 2n. For I={i1, . . . , ir} ⊂ {1, . . . , n}with 1≤i1< i2<· · ·< in≤n, put eI = ei1ei2· · ·eir, while forI=∅, e∅= e0. For 0≤r≤nfixed, the space C`rn is defined by
C`rn= span{eI :|I|:= card(I) =r}.
The Clifford algebras C`n is a graded algebra as C`n=⊕rC`rn.
Any elementa∈C`n may thus be written in a unique way as a= [a]0+ [a]1+· · ·+ [a]n
where [ ]r : C`n →C`rn denotes the projection of C`n onto C`rn. In particular, by C`2n =Hwe denote the algebra of real quaternion. It is customary to identifyR with C`0n and identifyRn with C`1n respectively. This means that each elementx ofRn may be represented by
x=
n
X
i=1
xiei.
Foru∈C`n, we denotes by [u]0 the scalar part of u, that is the coefficient of the element e0. We define the Clifford conjugation as follows:
ei1ei2· · ·eir = (−1)r(r+1)2 ei1ei2· · ·eir. We denote
(A, B) = AB
0.
Then an inner product is thus obtained, giving to the norm| · |on C`n given by
|A|2= AA
0.
From [15], we know that this norm is submultiplicative: |AB| ≤ C(n)|AkB|, whereC(n) is a positive constant depending only onnand no more than 2n/2.
In what follows, we let Rn+ ={(x1, . . . , xn)∈ Rn :xn >0} and Σ = ∂Rn+. A Clifford-valued function u:Rn+ →C`n can be written as u= ΣIuIeI, where the coefficientsuI :Rn+→Rare real-valued functions.
The Dirac operator on the Euclidean space used here is introduced by D=
n
X
j=1
ej
∂
∂xj :=
n
X
j=1
ej∂j.
Ifuis a real-valued function defined onRn+, thenDu=∇u= (∂1u, ∂2u, . . . , ∂nu).
Moreover, D2 = −∆, where ∆ is the Laplace operator which operates only on coefficients. A function is left monogenic if it satisfies the equationDu(x) = 0 for eachx∈Rn+. A similar definition can be given for right monogenic function. An important example of a left monogenic function is the generalized Cauchy kernel
G(x) = 1 ωn
x
|x|n,
whereωn denotes the surface area of the unit ball inRn. This function is a funda- mental solution of the Dirac operator. Basic properties of left monogenic functions one can refer to [11, 17] and references therein.
2.2. Variable exponent spaces of Clifford-valued functions. Next we recall some basic properties of variable exponent spaces of Clifford-valued functions. In what follows, we use the short notation Lp(x)(RN+), W1,p(x)(RN+), etc., instead of Lp(x)(RN+,R),W1,p(x)(RN+,R), etc. Throughout this paper we always assume (un- less declared specially)
p∈ Plog(Rn+)and1< p−:= inf
x∈Rn+
p(x)≤p(x)≤ sup
x∈Rn+
p(x) =:p+<∞. (2.1) where Plog(Rn+) is the set of exponent psatisfying the so-called log-H¨older conti- nuity, i.e.,
|p(x)−p(y)| ≤ C
log(e+|x−y|−1), |p(x)−p(∞)| ≤ C log(e+|x|−1)
hold for allx, y∈RN+, wherep(∞) = lim|x|→∞p(x), see [3, 5]. LetP(Rn+) be the set of all Lebesgue measurable functions p: Rn+ → (1,∞). Givenp∈ P(Rn+) we define the conjugate functionp0(x)∈ P(Rn+) by
p0(x) = p(x)
p(x)−1 for eachx∈RN+. The variable exponent Lebesgue spaceLp(x)(Rn+) is defined by
Lp(x)(Rn+) =
u∈ P(Rn+) : Z
Rn+
|u|p(x)dx <∞ ,
with the norm
kukLp(x)(Rn+)= inf t >0 :
Z
Rn+
|u
t|p(x)dx≤1 .
The variable exponent Sobolev spaceW1,p(x)(Rn+) in a half-space is defined by W1,p(x)(Rn+) =
u∈Lp(x)(Rn+) :|∇u| ∈Lp(x)(Rn+) , with the norm
kukW1,p(x)(Rn+)=k∇ukLp(x)(Rn+)+kukLp(x)(Rn+). (2.2) Denote W01,p(x)(Rn+) by the completion of C0∞(Rn+) in W1,p(x)(Rn+) with respect to the norm (2.2). The space W−1,p(x)(Rn+) is defined as the dual of the space W01,p0(x)(Rn+). For more details we refer to [3, 7, 8] and reference therein.
In the following, we say that u ∈ Lp(x)(Rn+,C`n) can be understood coordi- nate wise. For example, u ∈ Lp(x)(Rn+,C`n) means that {uI} ⊂ Lp(x)(Rn+) for u = ΣIuIeI ∈ C`n with the norm kukLp(x)(Rn+,C`n) = P
IkuIkLp(x)(Rn+). In the same way, spacesW1,p(x)(Rn+,C`n), W01,p(x)(Rn+,C`n),C0∞(Rn+,C`n), etc., can be understood similarly. In particular, the space L2(Rn+,C`n) can be converted into a right Hilbert C`n-module by defining the following Clifford-valued inner product (see [14, Definition 3.74])
f, g
C`n = Z
Rn+
f(x)g(x)dx. (2.3)
Remark 2.1. Following the same arguments as in [10, 37], we can calculate easily thatkukLp(x)(Rn+,C`n)is equivalent to the normk|u|kLp(x)(Rn+). Furthermore, we also prove that for everyu∈W01,p(x)(Rn+,C`n),kDukLp(x)(Rn+,C`n)is an equivalent norm ofkukW1,p(x)
0 (Rn+,C`n).
Lemma 2.2 ([10]). Assume thatp(x)∈ P(Rn+). Then Z
Rn+
|uv|dx≤C(n, p)kukLp(x)(Rn+,C`n)kvkLp0(x)(Rn+,C`n)
for everyu∈Lp(x)(Rn+,C`n)andv∈Lp0(x)(Rn+,C`n).
Lemma 2.3([10, 11]). Ifp(x)∈ P(Rn+), thenLp(x)(Rn+,C`n)andW1,p(x)(Rn+,C`n) are reflexive Banach spaces.
Based on the Cauchy kernel G(x) we can introduce the Teodorescu operator.
There exist a number of applications and methods based on the properties of this Teodorescu operator. But in our case of considering the domain Ω as an unbounded domain, the main problem in applying this operator is that the Cauchy kernel does not have good enough behaviour near infinity. For example, the Teodorescu operator is an unbounded operator over the usual function spaces on Ω. In this paper, we will follow the idea from [2, 18] by using add-on terms to the Cauchy kernel. More precisely, we choose a fixed pointz lying in the complement of RN+. Then we consider the following operators.
Definition 2.4 ([2, 14, 15]). (i) Letu∈C(Rn+,C`n). The Teodorescu operator is defined by
T u(x) = Z
Rn+
Kz(x, y)u(y)dy,
where Kz(x, y) = G(x−y)−G(y−z), G(x) is the above-mentioned generalized Cauchy kernel.
(ii) Letu∈C1(Rn+,C`n)∩C(Rn+,C`n). Theboundary operatoris defined by F u(x) =
Z
Σ
Kz(x, y)α(y)u(y)dSy, whereα(y) denotes the outward normal unit vector aty.
(iii) Letu∈L1loc(Rn). Then the Hardy-Littlewood maximal operator is defined by
M u x
= sup
x∈Q
1
|Q|
Z
Q
|u(y)|dy.
for allx∈Rn, where the supremum is taken over all cubes (or ball)Q⊂Rn which containx.
The Teodorescu operator was first introduced in [18] and the operator properties in the scale ofWk,2-spaces were given in [23], see also [2] for the corresponding op- erator properties in theWk,q-spaces over unbounded domains. Its main advantage is a faster decay to infinity of the kernel.
Lemma 2.5. (see [3])Let x∈Rn,δ >0andu∈L1loc(Rn). Then Z
B(x,δ)
1
|x−y|n−1|u(y)|dy≤C(δ)M u(x).
whereC(δ)>0is a positive constant. Moreover, ifu∈Lp(x)(Rn)withkukp(x)≤1,
then Z
Rn\B(x,δ)
1
|x−y|n−1|u(y)|dy≤C(n, p, δ,|B|).
whereC(n, p, δ,|B|)is a positive constant.
Lemma 2.6 ([3]). Let p(x)satisfy (2.1). ThenM is bounded inLp(x)(Rn).
Lemma 2.7 ([18]). Let u∈C1(Rn+,C`n). Then
∂kT u(x) = 1 ωn
Z
Rn+
∂
∂xk
G(x−y)u(y)dy+u(x) n ek.
Lemma 2.8([3]). LetΦbe a Calder´on-Zygmund operator with Calder´on-Zygmund kernelK onRn×Rn. Then Φis bounded onLp(x)(Rn).
Lemma 2.9. The following operators are continuous linear operators:
(i) T :Lp(x)(Rn+,C`n)→W1,p(x)(Rn+,C`n).
(ii) Te:W−1,p(x)(Rn+,C`n)→Lp(x)(Rn+,C`n).
Proof. (i) We divide the proof into two parts:
Step 1: The operator∂kT : Lp(x)(Rn+,C`n)→Lp(x)(Rn+,C`n) is continuous. By Lemma 2.7 we have foru∈C0∞(Rn+,C`n)
∂kTu(x) = 1 ωn
Z
Rn+
∂
∂xkKz(x, y)u(y)dy+u(x) n ek. LetK(x, y) = ω1
n
∂
∂xkKz(x, y). Since ω1
n
∂
∂xkKz(x, y) = ω1
n
∂
∂xkG(x−y) and
∂
∂xkG(x−y) = 1
|x−y|n
ek−n
n
X
i=1
(xk−yk)(xi−yi)
|x−y|2 ei
,
we obtain
∂
∂xk
G(x−y)
≤ n2+ 1
|x−y|n, (k= 1, . . . , n).
Notice that
Z
S1
ek−n
n
X
i=1
(xk−yk)(xi−yi)
|x−y|2 ei
dS= 0,
where S1={y∈Rn+:|x−y|= 1}. Then it is easy to verify thatK(x, y) satisfies the following properties:
(a) |K(x, y)| ≤C|x−y|−n; (b) K t(x, y)
=t−nK(x, y), t >0;
(c) R
S1K(x, y)dS= 0.
Now we defineu(x) = 0 for x∈Rn\Rn+. Then K(x, y) satisfies the conditions of Calder´on-Zygmund kernal onRn×Rn. By Theorem 2.13, we know the inequality can be extended toLp(x)(Rn+,C`n). Therefore, we obtain by Lemma 2.5 and Lemma 2.6
k 1 ωn
Z
Ω
∂k,xG(x−y)u(y)dykLp(x)(Rn+,C`n)≤C(n, p)kukLp(x)(Rn+,C`n) (2.4) On the other hand,
ku(x)
n ekkLp(x)(Rn+,C`n)≤ 1
nkukLp(x)(Rn+,C`n) (2.5) Combining (2.3) with (2.5), we obtain
k∂kTukLp(x)(Rn,Cln)≤C(n, p)kukLp(x)(Rn+,C`n).
Step 2: The operator T : Lp(x)(Rn+,C`n) → Lp(x)(Rn+,C`n) is continuous. We defineu(x) = 0 forx∈Rn\Rn+. Since
|G(x−y) +G(y−z)| ≤ 1 ωn
1
|x−y|n−1 + 1
|y−z|n−1
,
we have
|T u(x)| ≤C(n) Z
Rn+
(|G(x−y)|+|G(y−z)|)|u(y)|dy
≤CZ
Rn+
1
|x−y|n−1|u(y)|dy+ Z
Rn+
1
|y−z|n−1|u(y)|dy .
Then by Lemmas 2.5 and 2.6, we obtain
kT ukLp(x)(Rn+,C`n)≤C(n, p)kukLp(x)(Rn+,C`n). Finally, combining Step 1 with Step 2, we have
kT ukW1,p(x)(Rn+,C`n)=kT ukLp(x)(Rn+,C`n)+
n
X
k=1
k∂kTukLp(x)(Rn+,C`n)
≤C(n, p)kukLp(x)(Rn+,C`n). Then we obtain the desired conclusion (i).
(ii) In view of [3, Proposition 12.3.2], we know that for eachf ∈W−1,p(x)(Rn+), there existsfk∈Lp(x)(Rn+), k= 0,1, . . . , n, such that
hf, ϕi=
n
X
k=0
Z
Rn+
fk
∂ϕ
∂xkdx, (2.6)
for allϕ∈W1,p
0(x)
0 (Rn+). Moreover,kfkW−1,p(x)(Rn+)is equivalent toPn
k=0kfkkLp(x)(Rn+). Obviously, for every f ∈W−1,p(x)(Rn+,C`n) the equality (2.5) still holds for fk ∈ Lp(x)(Rn+,C`n), k = 0,1, . . . , n. Moreover, kfkW−1,p(x)(Rn+,C`n) is equivalent to Pn
k=0kfkkLp(x)(Rn+,C`n). On the other hand, by [3, Proposition 12.3.4], it is easy to show that the spaceC0∞(Rn+,C`n) is dense inW−1,p(x)(Rn+,C`n). Thus we may choose
uj =uj0+
n
X
k=1
∂ujk
∂xk
,
where uj0, ujk ∈ C0∞(Rn+,C`n), such that kuj−fkW−1,p(x)(Rn+,C`n) → 0 andkujk− fkkLp(x)(Rn+,C`n)→0 as j→ ∞, wherek= 0,1, . . . , n. Here, we are using the fact thatC0∞(Rn+,C`n) is dense inLp(x)(Rn+,C`n)(see [3]). Then we consider
T uj= Z
Rn+
Kz(x, y)uj(y)dy.
Then we have T uj =
Z
Rn+
Kz(x, y) uj0(y) +
n
X
k=1
∂
∂yk
ujk(y) dy
= Z
Rn+
Kz(x, y)uj0(y)dy−
n
X
k=1
Z
Rn+
∂
∂yk
Kz(x, y)ujk(y)dy.
Since Z
Rn+
Kz(x, y)uj0(y)dy ≤
Z
Rn+
1
|x−y|n−1 uj0(y)
dy+ Z
Rn+
1
|y−z|n−1 uj0(y)
dy.
By Remark 2.1, Lemma 2.5 and Lemma 2.6, there exists a constant C0 >0 such that
Z
Rn+
Kz(x, y)uj0(y)dy Lp(x)(
Rn+,C`n)≤C0kuj0kLp(x)(Rn+,C`n). (2.7)
Now let us extendujk(x) by zero toRn\Rn+. Note that the position ofz which is outside of a half space Rn+ leads to the fact that G(y−z) has no singularities for any y ∈Rn+. Thus it is easy to show that ∂y∂
kKz(x, y) satisfies the conditions of Calder´on-Zygmund kernel onRn×Rn. In view of Lemma 2.8, there exist positive constantCk(k= 1, . . . , n) such that
Z
Rn+
∂
∂ykKz(x, y)ujk(y) Lp(x)(
Rn+,C`n)≤CkkujkkLp(x)(Rn+,C`n). (2.8) Combining (2.7) with (2.8), we have
kT ujkLp(x)(Rn+,C`n)≤C0kuj0kLp(x)(Rn+,C`n)+
n
X
k=1
CkkujkkLp(x)(Rn+,C`n).
Lettingj→ ∞, by means of the Continuous Linear Extension Theorem, the oper- atorT can be uniquely extended to a bounded linear operatorTesuch that for all f ∈W−1,p(x)(Rn+,C`n), there exists a constantC >e 0 such that
kT fe kLp(x)(Rn+,C`n)≤C
kf0kLp(x)(Rn+,C`n)+
n
X
k=1
kfkkLp(x)(Rn+,C`n)
≤Ckfe kW−1,p(x)(Rn+,C`n).
Hence claim (ii) follows.
Lemma 2.10. The following operators are continuous linear operators:
(i) D:W1,p(x)(Rn+,C`n)→Lp(x)(Rn+,C`n).
(ii) De :Lp(x)(Rn+,C`n)→W−1,p(x)(Rn+,C`n).
Proof. (i) The proof is similar to that of [11, Lemma 2.6], so we omit it.
(ii) We consider the following Dirichlet problem of the Poisson equation with homogeneous boundary data
−∆u=f, in Rn+
u= 0, on Σ (2.9)
It is easy to see that for allf ∈W−1,p(x)(Rn+,C`n) problem (2.9) still has a unique weak solution u ∈ W1,p(x)(Rn+,C`n), see Diening, Lengeler and Ruˇziˇcka [5]. We denote by ∆−10 the solution operator. On the other hand, it is clear that the operator
∆ :W1,p(x)(Rn+,C`n)→W−1,p(x)(Rn+,C`n)
is continuous, so we obtain from Lemma 2.9 that the operator De = −∆T : Lp(x)(Rn+,C`n)→W−1,p(x)(Rn+,C`n) is continuous, where the operatorDe can be considered as a unique continuous linear extension of the operatorD.
Lemma 2.11. Let p(x)∈ P(Rn+).
(i) Ifu∈W1,p(x)(Rn+,C`n), then the Borel-Pompeiu formulaF u(x)+T Du(x) = u(x)holds for allx∈Rn+.
(ii) If u ∈ Lp(x)(Rn+,C`n), then the equation DT u(x) = u(x) holds for all x∈Rn+.
Proof. Let us denote by C0∞(Rn+) the space of all restrictions of functions from C0∞(Rn) toRn+. Furthermore, suppose ϕ∈C0∞(Rn+). Now, let us consider a point y ∈Ω and the open ball B(0, r) with origin 0, radius r, and boundary S(0, r). If ris sufficiently large such that y lies in the domain Ω(r) =B(0, r)∩Rn+. For this domain, we have
FS(0,r)ϕ(y) =ϕ(y)−TΩ(r)Dϕ(y).
see [25] for more details. This can be written in the form
r→∞lim Z
Σ∩B(0,r)
+ Z
S(0,r)∩Rn+
Kz(x, y)α(y)u(y)dSy
=ϕ(y)− lim
r→∞TΩ(r)Dϕ(y) Since
r→∞lim Z
Σ∩B(0,r)
Kz(x, y)α(y)u(y)dSy= Z
Σ
Kz(x, y)α(y)u(y)dSy
and
r→∞lim TΩ(r)Dϕ(y) =TRn
+Dϕ(y), lim
r→∞
Z
S(0,r)∩Rn+
Kz(x, y)α(y)u(y)dSy = 0, we obtain the Borel-Pompeiu formula in case ofϕ∈C0∞(Rn+). Finally, the desired result (i) follows immediately from the density document.
(ii) Using the same idea with (i), we can get directly the desired result from [25,
Lemma 2.6].
Lemma 2.12. Let p(x) satisfy (2.1).
(i) If u∈Lp(x)(Rn+,C`n), thenTeDu(x) =e u(x)for all x∈Rn+. (ii) If u∈W−1,p(x)(Rn+,C`n), thenDeT u(x) =e u(x)for allx∈Rn+.
Proof. (i) follows from Lemma 2.11 (i) and the denseness of W01,p(x)(Rn+,C`n) in Lp(x)(Rn+,C`n).
(ii) follows from Lemma 2.11 (ii) and the denseness ofC0∞(Rn+,C`n) in the space W−1,p(x)(Rn+,C`n), see [3, Proposition 12.3.4] for the details.
G¨urlebeck and Spr¨oßig [14, 15] showed that the orthogonal decomposition of the spaceL2(Ω) holds in the hyper-complex function theory:
L2(Ω,C`n) = (kerD∩L2(Ω,C`n))⊕DW01,2(Ω,C`n) (2.10) with respect to the Clifford-valued product (2.3). Note that kerD denotes the set of all monogenic functions on Ω. This decomposition has a number of applications, especially to the theory of partial differential equations, see [6] for the complex case and [14] for the hyper-complex case. K¨ahler [22] extended the orthogonal decomposition (2.10) to the spacesLp(Ω) in form of a direct decomposition in the case of Clifford analysis. In [7], Fu et al. extended the direct decomposition to the case of variable exponent Lebesgue spaces in bounded smooth domains.
Theorem 2.13. The spaceLp(x)(Rn+,C`n)allows the Hodge-type decomposition Lp(x)(Rn+,C`n) = (kerDe∩Lp(x)(Rn+,C`n))⊕DW01,p(x)(Rn+,C`n) (2.11) with respect to the Clifford-valued product (2.3).
Proof. Similar to the proof of [22, Theorem 6], we first show that the intersection of (kerDe ∩Lp(x)(Rn+,C`n)) and DW01,p(x)(Rn+,C`n) is empty. Suppose f belongs to both kerDe∩Lp(x)(Rn+,C`n) andDW01,p(x)(Rn+,C`n), thenDfe = 0 . Becausef belongs toDW01,p(x)(Rn+,C`n), there exists a function v ∈ W01,p(x)(Rn+,C`n) such that Dv = f. Hence, we obtain that −∆v = 0 and v = 0 on Σ. From the uniqueness of ∆−10 we obtainv = 0. Consequently, f = 0. Therefore, the sum of the two subspaces is a direct one.
Now letu∈ Lp(x)(Rn+,C`n). Then u2 =D∆−10 Due ∈DW01,p(x)(Rn+,C`n). Let u1=u−u2. Thenu1∈Lp(x)(Rn+,C`n). Furthermore, we takeuk ∈W01,p(x)(Rn+,C`n) such thatuk→uinLp(x)(Rn+,C`n), then by the denseness ofW01,p(x)(Rn+,C`n) in Lp(x)(Rn+,C`n) and Lemma 2.2, we have for anyϕ∈W01,p0(x)(Rn+,C`n)
u1, Dϕ
C`n= u−u2, Dϕ
C`n
= lim
k→∞ Duk−DD∆−10 Duk, ϕ
C`n
= lim
k→∞ Duk−Duk, ϕ
C`n= 0,
which implies that u1 ∈kerD. Sincee u∈Lp(x)(Rn+,C`n) is arbitrary, the desired
result follows.
From this decomposition we can get the following two projections P :Lp(x)(Rn+,C`n)→kerDe∩Lp(x)(Rn+,C`n),
Q:Lp(x)(Rn+,C`n)→DW01,p(x)(Rn+,C`n).
Moreover, we have
Q=D∆−10 D, Pe =I−Q.
Corollary 2.14. The spaceLp(x)(Rn+,C`n)∩imQis a closed subspace ofLp(x)(Rn+,C`n).
The proof can be easily done by combining Theorem 2.13, Lemma 2.3 with Lemma 2.10. We refer the reader to [38, Lemma 2.6] for a similar argument.
Corollary 2.15. Lp(x)(Rn+,C`n)∩imQ∗
=Lp0(x)(Rn+,C`n)∩imQ. Namely, the linear operator
Φ :DW1,p
0(x)
0 (Rn+,C`n)→ DW01,p(x)(Rn+,C`n)∗ given by
Φ(Du)(Dϕ) = (Dϕ, Du)Sc:=
Z
Rn+
DϕDu
0dx is a Banach space isomorphism.
Proof. In terms of Lemma 2.14, DW01,p(x)(Rn+,C`n) and DW1,p
0(x)
0 (Rn+,C`n) are reflexive Banach spaces since they are closed subspaces in Lp(x)(Rn+,C`n) and Lp0(x)(Rn+,C`n) respectively. The linearity of Φ is clear. For injectivity, suppose
Φ(Du)(Dϕ) = (Dϕ, Du)Sc= 0 (2.12)
for all ϕ ∈ W01,p(x)(Rn+,C`n) and some u ∈ W01,p0(x)(Rn+,C`n). For any ω ∈ Lp(x)(Rn+,C`n), according to (2.11), we may write ω = α+β with α ∈ kerDe ∩
Lp(x)(Rn+,C`n) andβ ∈DW01,p(x)(Rn+,C`n). Thus we obtain (ω, Du)Sc= (α+β, Du)Sc= (β, Du)Sc.
This and (2.12) gives (ω, Du)Sc= 0. This means thatDu= 0. It follows that Φ is injective. To get subjectivity, letf ∈ DW01,p(x)(Rn+,C`n)∗
. By the Hahn-Banach Theorem, there isF ∈ Lp(x)(Rn+,C`n)∗
withkFk=kfkandF|DW1,p(x)
0 (Rn+,C`n)= f. Moreover, there existsϕ∈Lp0(x)(Rn+,C`n) such thatF(u) = (u, ϕ)Scfor anyu∈ Lp(x)(Rn+,C`n). According to (2.11), we can writeϕ=η+Dα, whereη∈kerDe∩ Lp0(x)(Rn+,C`n),Dα∈DW01,p0(x)(Rn+,C`n). For anyDu∈DW01,p(x)(Rn+,C`n), we have
f(Du) = (Du, ϕ)Sc= (Du, Dα)Sc= Φ(Dα)(Du).
Consequently, Φ(Dα) =f. It follows that Φ is surjective. By [10, Theorem 3.1] we have
|Φ(Du)(Dϕ)| ≤CkDϕkLp(x)(Rn+,C`n)kDukLp0(x)(Rn+,C`n).
This means that Φ is continuous. Furthermore, it is immediate that Φ−1is contin- uous from the Inverse Function Theorem. This ends the proof of Lemma 2.3.
3. Stokes equations in the half-space
In the section, we consider the Stokes system which consists in finding a solution (u, π) for
−∆u+ 1
µ∇π= ρ
µf inRn+, (3.1)
divu=f0 in Rn+, (3.2)
u=v0 on Σ. (3.3)
With R
Ωf0dx = R
∂Ωn·v0dx the necessary condition for the solvability is given.
Here, uis the velocity, π the hydrostatic pressure, ρ the density, µ the viscosity, f the vector of the external forces and the scalar function f0 a measure of the compressibility of fluid. The boundary condition (3.3) describes the adhesion at the boundary of the domain Ω for v0 = 0. This system describes the stationary flow of a homogeneous viscous fluid for small Reynold’s numbers. For more details, we refer to [2, 14, 15, 16, 20].
In this paper, for f =Pn
i=1fiei and u=Pn
i=1uiei, we consider the following Stokes system in the hyper-complex formulation (see [16, 17]):
DDue +1
µDπ= ρ
µf in Rn+, (3.4)
[Du]0= 0 in Rn+, (3.5)
u= 0 on Σ. (3.6)
Definition 3.1. We say that (u, π) ∈ W01,p(x)(Rn+,C`n)×Lp(x)(Rn+) is a solu- tion of (3.4)–(3.6) provided that it satisfies the system (3.4)–(3.6) for all f ∈ W−1,p(x)(Rn+,C`n).
Definition 3.2. The operator∇e :Lp(x)(Rn+)→(W−1,p(x)(Rn+))n is defined by h∇f, ϕie =−hf,divϕi:=−
Z
Rn+
fdivϕdx for allf ∈Lp(x)(Rn+) andϕ∈(C0∞(Rn+))n.
Theorem 3.3. Suppose f ∈ W−1,p(x)(Rn+,C`n). Then the Stokes system (3.4)–
(3.6)has a unique solution(u, π)∈W01,p(x)(Rn+,C`n)×Lp(x)(Rn+) of the form u+ 1
µT Qπ= ρ µT QT f,e with respect to the estimate
kDukLp(x)(Rn+,C`n)+ 1
µkQπkLp(x)(Rn+)≤Cρ
µkQT fe kLp(x)(Rn+,C`n).
Here,C≥1is a constant and the hydrostatic pressureπis unique up to a constant.
Proof. We first prove that iff ∈W−1,p(x)(Rn+,C`n), then we have the representa- tion
T QT fe =u+T Qω.
Indeed, let ϕn ∈ W01,p(x)(Rn+,C`n) with ϕn → ϕ in Lp(x)(Rn+,C`n). By Lemma 2.11, we have
T QT(Dϕn) =T Qϕn.
SinceW01,p(x)(Rn+,C`n) is dense inLp(x)(Rn+,C`n), it follows thatT QTeDϕe =T Qϕ.
Thus, foru∈W01,p(x)(Rn+,C`n) andπ∈Lp(x)(Rn+) we obtain T QTe(ρ
µf) =T QTe(DDue + 1
µDπ) =e u+ 1 µT Qπ.
This implies that our system (3.4)–(3.5) is equivalent to the system u+ 1
µQT Qπ= ρ
µT QT f,e (3.7)
[Qπ]0= [QT f]e 0. (3.8)
Obviously, the equality (3.4) is equivalent to the equality Du+ 1
µQπ= ρ
µQT f.e (3.9)
Now we need to show that for each f ∈ W−1,p(x)(Rn+,C`1n), the function QT f can be decomposed into two functions Du and Qπ. Suppose Du+Qπ = 0 for u ∈ W01,p(x)(Rn+,C`1n)∩ker div and π ∈ Lp(x)(Rn+). Then (3.5) gives [Qπ]0 = 0.
Thus,Qπ = 0. Hence, Du=Qπ = 0. This means thatDu+Qπ is a direct sum, which is a subset of imQ.
Next we have to ask about the existence of a functionalF ∈(Lp(x)(Rn+,C`1n)∩ imQ)∗ with F(Du) = 0 andF(Qπ) = 0 but F(QT f)e 6= 0. This is equivalent to ask if there existsg∈W−1,p0(x)(Rn+,C`1n), such that for allu∈W01,p(x)(Rn+,C`1n)∩ ker div andω∈Lp(x)(Rn+),
(Du, QT g)e Sc= 0, (3.10)
(Qπ, QT g)e Sc= 0, (3.11)
but (QT f, Qe T g)e Sc6= 0. Here, Lemmas 2.9 and Corollary 2.15 are employed.
Thus, let us consider the system (3.10) and (3.11) with g ∈ W−1,p0(x)(Ω,C`1n) for all open cubes Ω⊂Rn+. Notice that, with the help of Lemma 2.10, (3.10) yields
(Du, QT g)e Sc = (u,DQe T g)e Sc= (u,DeT ge −DPe T g)e Sc = (u, g)Sc= 0, which implies g = ∇he = Dhe with h ∈ Lploc0(x)(Rn+) because of [38, Lemma 2.8].
Furthermore, by Definition 3.2, it is easy to see that if g ∈ W−1,p0(x)(Rn+,C`1n), thenh∈Lp0(x)(Rn+). Thus it follows from (3.11) and Lemma 2.3,
(Qπ, QT g)e Sc= (Qπ, QTeDh)e Sc= (Qπ, Qh)Sc= 0
holds for eachπ∈Lp(x)(Rn+). Hence, Qπ=|Qh|p0(x)−2QhgivesQh= 0. Then we obtain
g=Dhe =DQhe +DP he = 0.
Furthermore, we obtain
(QT f, Qe T g)e Sc= 0, for allf ∈W−1,p(x)(Rn+,C`1n).
Finally, (3.9) yields
kDukLp(x)(Rn+,C`n)+ 1
µkQπkLp(x)(Rn+)≥ ρ
µkQT fe kLp(x)(Rn+,C`n). By the Norm Equivalence Theorem, we obtain
kDukLp(x)(Rn+,C`n)+ 1
µkQπkLp(x)(Rn+)≤Cρ
µkQT fe kLp(x)(Rn+,C`n). By Remark 2.1, Lemma 2.9 and the boundedness of the operatorQ, we obtain
kukW1,p(x)
0 (Rn+,C`n)+ 1
µkQπkLp(x)(Rn+)≤Cρ
µkfkW−1,p(x)(Rn+,C`n), (3.12) which implies the uniqueness of solution. Note that Qπ = 0 implies π ∈ kerD.e Therefore,πis unique up to a constant. The proof is complete.
4. N-S equations in the half-space
In this section, we consider the time-independent Navier-Stokes equations in variable exponent spaces of Clifford-valued functions in a half-space:
−∆u+ρ
µ(u· ∇)u+1
µ∇π= ρ
µf inRn+, (4.1)
divu=f0 in Rn+, (4.2)
u=v0 on Σ. (4.3)
In addition to the case of the Stokes system, the main difference from the above- mentioned Stokes equations is the appearance of the non-linear convection term (u· ∇)u. In 1928, Oseen showed that one can get relatively good results if the convection term (u· ∇)u is replaced by (v · ∇)u , where v is a solution of the corresponding Stokes equations. In 1965, Finn [13] proved the existence of solutions for small external forces with a spatial decreasing to infinity of order|x|−1 for the case ofn= 3, and used the Banach fixed-pointed theorem. G¨urlebeck and Spr¨oßig [14, 15, 17] solved this system by a reduction to a sequence of Stokes problems provided the external force f belongs to Lp(Ω,H) for a bounded domain Ω and 6/5 < p < 3/2. Cerejeiras and K¨ahler [2] obtained the similar results provided
